arXiv subjects

Salvador Hernández

Publications and source records attributed to Salvador Hernández.

At least 19 recordsLinked to original sources

arXiv2026

Duality and Baire like properties in topological abelian groups

This article investigates the duality of abelian topological groups by delving into the interplay between the algebraic and topological properties of a group $G$ and those of its dual group. In particular, extending the notion of a $g$-barrelled group, we introduce and study the classes of $κ$-barrelled and $ω$-barrelled groups. Our first main result establishes that every $ω$-barrelled group is $\aleph_0$-barrelled (i.e., if every weakly convergent sequence in the dual group is equicontinuous, then every metrizable weakly compact subset of the dual group is also equicontinuous). From this, we deduce that the free topological abelian group $A(K)$ over a metrizable compact space $K$ is determined by the subgroup generated by any of its dense subsets. Furthermore, we prove that if $G$ is a $σ$-compact, $ω$-barrelled group, then its dual group $\widehat{G}_κ$ equipped with the compact-open topology is sequentially complete (complete if $G$ is hemicompact). We also solve an open question posed by Trigos-Arrieta by constructing an explicit example of an $ω$-barrelled metrizable group that fails to be $g$-barrelled. Finally, we investigate the dualization of groups lacking infinite compact subsets, proving that for a totally bounded abelian group $G$, every compact subset of $G$ is finite if and only if its dual group $\widehat{G}$, endowed with the finite-open topology, is unordered Baire-like.

math.FA↗

arXiv2025

Extension of continuous functions on product spaces, Bohr Compactification and Almost Periodic Functions

The Bohr compactification is a well known construction for (topological) groups and semigroups. Recently, this notion has been investigated for arbitrary structures in \cite{har_kun:bohr_discrete} where the Bohr compactification is defined, using a set-theoretical approach, as the maximal compactification which is compatible with the structure involved. Here, we give a characterization of the continuous functions defined on a product space that can be extended continuously to certain compactifications of the product space. As a consequence, the Bohr compactification of an arbitrary topological structure is obtained as the Gelfand space of the commutative Banach algebra of all almost periodic functions. Previously, almost periodic functions $f$ are defined in terms of translates of $f$ with no reference to any compactification of the underlying structure. An application is given to the representation of isometries defined between spaces of almost periodic functions.

math.FA↗

arXiv2025

Group homomorphisms induced by isometries

Let $G$ and $H$ be locally compact groups and consider their associate spaces of almost periodic functions $AP(G)$ and $AP(H)$. We investigate the continuous group homomorphisms induced by isometries of $AP(G)$ into $AP(H)$. Among others, the following results are proved: {\bf Theorem} Let $G$ and $H$ be $σ$-compact maximally almost periodic locally compact groups. Suppose that $T$ is a non-vanishing linear isometry of $AP(G)$ into $AP(H)$ that respects finite dimensional unitary representations. Then there is a closed subgroup $H_0\subseteq H$, a continuous group homomorphism $t$ of $H_0$ onto $G$ and an character $γ\in \widehat{H}$ such that $(Tf)(h)=γ(h)~f(t(h))$ for all $h\in H_0$ and for all $f\in C(G)$. {\bf Theorem} Let $G$ and $H$ be $LC$ Abelian groups and $H$ is connected. Suppose that $T$ is a non-vanishing linear isometry of $AP(G)$ into $AP(H)$ that preserves trigonometric polynomials. Then there is a closed subgroup $H_0\subseteq H$, a continuous group homomorphism $t$ of $H_0$ onto $G$, an element $h_0\in H_0$, a character $α\in \widehat{H}$ and an unimodular complex number $a$ such that $(Tf)(h)=a\cdot α(h)~\cdot f(t(h-h_0))\text{ for all }h\in H_0\text{ and for all }f\in C(G)\text{.}$

math.FA↗

arXiv2024

Tensor products of topological abelian groups and Pontryagin duality

Let $G$ be the group of all $\ZZ$-valued homomorphisms of the Baer-Specker group $\ZZ^\NN$. The group $G$ is algebraically isomorphic to $\ZZ^{(\NN)}$, the infinite direct sum of the group of integers, and equipped with the topology of pointwise convergence on $\ZZ^\NN$, becomes a non reflexive prodiscrete group. It was an open question to find its dual group $\hat{G}$. Here, we answer this question by proving that $\hat{G}$ is topologically isomorphic to $\ZZ^\NN\otimes_\mathcal{Q}\TT$, the (locally quasi-convex) tensor product of $\ZZ^\NN$ and $\TT$. Furthermore, we investigate the reflexivity properties of the groups of $C_p(X,\ZZ)$, the group of all $\ZZ$-valued continuous functions on $X$ equipped with the pointwise convergence topology, and $A_p(X)$, the free abelian group on a $0$-dimensional space $X$ equipped with the topology $t_p(C(X,\ZZ))$ of pointwise convergence topology on $C(X,\ZZ)$. In particular, we prove that $\hat{A_p(X)}\simeq C_p(X,\ZZ)\otimes_\mathcal{Q}\TT$ and we establish the existence of $0$-dimensional spaces $X$ such that $C_p(X,\ZZ)$ is Pontryagin reflexive.

math.FA↗

arXiv2022

On closed subgroups of precompact groups

It is a Theorem of W.~ W. Comfort and K.~ A. Ross that if $G$ is a subgroup of a compact Abelian group, and $S$ denotes those continuous homomorphisms from $G$ to the one-dimensional torus, then the topology on $G$ is the initial topology given by $S$. {Assume that $H$ is a subgroup of $G$. We study how} the choice of $S$ affects the topological placement and properties of $H$ in $G$. Among other results, we have {made significant} progress toward the solution of the following specific questions: How many totally bounded group topologies does $G$ admit such that $H$ is a closed (dense) subgroup? If $C_S$ denotes the poset of all subgroups of $G$ that are $S$-closed, ordered by inclusion, does $C_S$ has a greatest (resp. smallest) element? We say that a totally bounded (topological, resp.) group is an \textit{SC-group} (\textit{topologically simple}, resp.) if all its subgroups are closed (if $G$ and $\{e\}$ are its only possible closed normal subgroups, resp.) {In addition, we investigate the following questions.} How many SC-(topologically simple totally bounded, resp.) group topologies does an arbitrary Abelian group $G$ admit?

math.GR↗

arXiv2021

Homomorphic encoders of profinite abelian groups I

Let $\{G_i :i\in\N\}$ be a family of finite Abelian groups. We say that a subgroup $G\leq \prod\limits_{i\in \N}G_i$ is \emph{order controllable} if for every $i\in \mathbb{N}$ there is $n_i\in \mathbb{N}$ such that for each $c\in G$, there exists $c_1\in G$ satisfying that $c_{1|[1,i]}=c_{|[1,i]}$, $supp (c_1)\subseteq [1,n_i]$, and order$(c_1)$ divides order$(c_{|[1,n_i]})$. In this paper we investigate the structure of order controllable subgroups. It is proved that every order controllable, profinite, abelian group contains a subset $\{g_n : n\in\N\}$ that topologically generates the group and whose elements $g_n$ all have finite support. As a consequence, sufficient conditions are obtained that allow us to encode, by means of a topological group isomorphism, order controllable profinite abelian groups. Some applications of these results to group codes will appear subsequently \cite{FH:2021}.

math.GR↗

arXiv2021

Homomorphic encoders of profinite abelian groups II

Let $\{G_i :i\in\N\}$ be a family of finite Abelian groups. We say that a subgroup $G\leq \prod\limits_{i\in \N}G_i$ is \emph{order controllable} if for every $i\in \mathbb{N}$ there is $n_i\in \mathbb{N}$ such that for each $c\in G$, there exists $c_1\in G$ satisfying that $c_{1|[1,i]}=c_{|[1,i]}$, $supp (c_1)\subseteq [1,n_i]$, and order$(c_1)$ divides order$(c_{|[1,n_i]})$. In this paper we investigate the structure of order controllable group codes. It is proved that if $G$ is an order controllable, shift invariant, group code over a finite abelian group $H$, then $G$ possesses a finite canonical generating set. Furthermore, our construction also yields that $G$ is algebraically conjugate to a full group shift.

math.GN↗

arXiv2021

The weak compactification of locally compact groups

We further investigate the weak topology generated by the irreducible unitary representations of a group $G$. A deep result due to Ernest \cite{Ernest1971} and Hughes \cite{Hughes1973} asserts that every weakly compact subset of a locally compact (LC) group $G$ is compact in the LC-topology, generalizing thereby a previous result of Glicksberg \cite{glicks1962} for abelian locally compact (LCA) groups. Here, we first survey some recent findings on the weak topology and establish some new results about the preservation of several compact-like properties when going from the weak topology to the original topology of LC groups. Among others, we deal with the preservation of countably compactness, pseudocompactness and functional boundedness.

math.GN↗

arXiv2018

Representation of Group Isomorphisms I

Let $G$ be a metric group and let $\sA ut(G)$ denote the automorphism group of $G$. If $\sA$ and $\sB$ are groups of $G$-valued maps defined on the sets $X$ and $Y$, respectively, we say that $\sA$ and $\sB$ are \emph{equivalent} if there is a group isomorphism $H\colon\sA\to\sB$ such that there is a bijective map $h\colon Y\to X$ and a map $w\colon Y\to \sA ut (G)$ satisfying $Hf(y)=w[y](f(h(y)))$ for all $y\in Y$ and $f\in \sA$. In this case, we say that $H$ is represented as a \emph{weighted composition operator}. A group isomorphism $H$ defined between $\sA$ and $\sB$ is called \emph{separating} when for each pair of maps $f,g\in \sA$ satisfying that $f^{-1}(e_G)\cup g^{-1}(e_G)=X$, it holds that $(Hf)^{-1}(e_G)\cup (Hg)^{-1}(e_G)=Y$. Our main result establishes that under some mild conditions, every separating group isomorphism can be represented as a weighted composition operator. As a consequence we establish the equivalence of two function groups if there is a biseparating isomorphism defined between them.

math.GN↗

arXiv2018

Epireflections in topological algebraic structures

Let $\sR$ be an epireflective category of $\topo$ and let $F_\sR$\, be the epireflective functor associated with $\sR$. If $\sA$ denotes a (semi)topological algebraic subcategory of $\topo$, we study when $F_\sR\,(A)$ is an epireflective subcategory of $\sA$. We prove that this is always the case for semi-topological structures and we find some sufficient conditions for topological algebraic structures. We also study when the epireflective functor preserves products, subspaces and other properties. In particular, we solve an open question about the coincidence of epireflections proposed by Echi and Lazar in \cite[Question 1.6]{Echi:MPRIA} and repeated in \cite[Question 1.9]{Echi:TP}. Finally, we apply our results in different specific topological algebraic structures.

math.GN↗

arXiv2018

On the structure of abelian profinite groups

A subgroup $G$ of a product $\prod\limits_{i\in\mathbb{N}}G_i$ is \emph{rectangular} if there are subgroups $H_i$ of $G_i$ such that $G=\prod\limits_{i\in\mathbb{N}}H_i$. We say that $G$ is \emph{weakly rectangular} if there are finite subsets $F_i\subseteq \mathbb{N}$ and subgroups $H_i$ of $\bigoplus\limits_{j\in F_i} G_j$ that satisfy $G=\prod\limits_{i\in\mathbb{N}}H_i$. %We say that $G$ is a \emph{subdirect product} of the family $\{G_i\}_{i\in I}$ if $G$ is weakly rectangular and %$G\cap\bigoplus\limits_{i\in I} G_i=\bigoplus\limits_{i\in\mathbb{N}}H_i$. In this paper we discuss when a closed subgroup of a product is weakly rectangular. Some possible applications to the theory of group codes are also highlighted.

math.GR↗

arXiv2018

When a totally bounded group topology is the Bohr Topology of a LCA group

We look at the Bohr topology of maximally almost periodic groups (MAP, for short). Among other results, we investigate when a totally bounded abelian group $(G,w)$ is the Bohr reflection of a locally compact abelian group. Necessary and sufficient conditions are established in terms of the inner properties of $w$. As an application, an example of a MAP group $(G,t)$ is given such that every closed, metrizable subgroup $N$ of $bG$ with $N \cap G = \{0\}$ preserves compactness but $(G,t)$ does not strongly respects compactness. Thereby, we respond to Questions 4.1 and 4.3 in [comftrigwu].

math.GN↗

arXiv2018

A dichotomy property for locally compact groups

We extend to metrizable locally compact groups Rosenthal's theorem describing those Banach spaces containing no copy of $l_1$. For that purpose, we transfer to general locally compact groups the notion of interpolation ($I_0$) set, which was defined by Hartman and Ryll-Nardzewsky [25] for locally compact abelian groups. Thus we prove that for every sequence $\lbrace g_n \rbrace_{n<ω}$ in a locally compact group $G$, then either $\lbrace g_n \rbrace_{n<ω}$ has a weak Cauchy subsequence or contains a subsequence that is an $I_0$ set. This result is subsequently applied to obtain sufficient conditions for the existence of Sidon sets in a locally compact group $G$, an old question that remains open since 1974 (see [32] and [20]). Finally, we show that every locally compact group strongly respects compactness extending thereby a result by Comfort, Trigos-Arrieta, and Wu [13], who established this property for abelian locally compact groups.

math.GN↗

arXiv2018

Interpolation sets in spaces of continuous metric-valued functions

Let $X$ and $M$ be a topological space and metric space, respectively. If $C(X,M)$ denotes the set of all continuous functions from X to M, we say that a subset $Y$ of $X$ is an \emph{$M$-interpolation set} if given any function $g\in M^Y$ with relatively compact range in $M$, there exists a map $f\in C(X,M)$ such that $f_{|Y}=g$. In this paper, motivated by a result of Bourgain in \cite{Bourgain1977}, we introduce a property, stronger than the mere \emph{non equicontinuity} of a family of continuous functions, that isolates a crucial fact for the existence of interpolation sets in fairly general settings. As a consequence, we establish the existence of $I_0$ sets in every nonprecompact subset of a abelian locally $k_ω$-groups. This implies that abelian locally $k_ω$-groups strongly respects compactness.

math.GN↗

arXiv2017

A classification of the cofinal structures of precompacta

We provide a complete classification of the possible cofinal structures of the families of precompact (totally bounded) sets in general metric spaces, and compact sets in general complete metric spaces. Using this classification, we classify the cofinal structure of local bases in the groups $\C(X,\bbR)$ of continuous real-valued functions on complete metric spaces $X$, with respect to the compact-open topology.

math.GN↗

arXiv2015

A countable free closed non-reflexive subgroup of Z^c

We prove that the group G=Hom(P,Z) of all homomorphisms from the Baer-Specker group P to the group Z of integer numbers endowed with the topology of pointwise convergence contains no infinite compact subsets. We deduce from this fact that the second Pontryagin dual of G is discrete. As G is non-discrete, it is not reflexive. Since G can be viewed as a closed subgroup of the Tychonoff product of continuum many copies of the integers Z, this provides an example of a group described in the title, thereby answering Problem 11 from [J.Galindo, L.Recorder-Núñez, M.Tkachenko, Reflexivity of prodiscrete topological groups, J. Math. Anal. Appl. 384 (2011), 320--330.] It follows that an inverse limit of finitely generated (torsion-)free discrete abelian groups need not be reflexive.

math.GN↗

arXiv2014

Representation of group isomorphisms. The compact case

Let $G$ be a discrete group and let $\mathcal A$ and $\mathcal B$ be two subgroups of $G$-valued continuous functions defined on two $0$-dimensional compact spaces $X$ and $Y$. A group isomorphism $H$ defined between $\mathcal A$ and $\mathcal B$ is called \textit{separating} when for each pair of maps $f,g\in \mathcal A$ satisfying that $f^{-1}(e_G)\cup g^{-1}(e_G)=X$, it holds that $Hf^{-1}(e_G)\cup Hg^{-1}(e_G)=Y$. We prove that under some mild conditions every separating isomorphism $H:\mathcal A\longrightarrow \mathcal B$ can be represented by means of a continuous function $h: Y\longrightarrow X$ as a weighted composition operator. As a consequence we establish the equivalence of two subgroups of continuous functions if there is a biseparating isomorphism defined between them.

math.GN↗

arXiv2014

Nonmeasurable subgroups of compact groups

In 1985 S.~Saeki and K.~Stromberg published the following question: {\it Does every infinite compact group have a subgroup which is not Haar measurable?} An affirmative answer is given for all compact groups with the exception of some metric profinite groups known as strongly complete. In this spirit it is also shown that every compact group contains a non-Borel subgroup.

math.GR↗